348 research outputs found
Gromov-Witten classes, quantum cohomology, and enumerative geometry
The paper is devoted to the mathematical aspects of topological quantum field
theory and its applications to enumerative problems of algebraic geometry. In
particular, it contains an axiomatic treatment of Gromov-Witten classes, and a
discussion of their properties for Fano varieties. Cohomological Field Theories
are defined, and it is proved that tree level theories are determined by their
correlation functions. Applications to counting rational curves on del Pezzo
surfaces and projective spaces are given.Comment: 44 p, amste
Influence of gravitational forces and fluid flows on a shape of surfaces of a viscous fluid of capillary size
The Navier-Stokes equations and boundary conditions for viscous fluids of
capillary size are formulated in curvilinear coordinates associated with a
geometry of the fluid-gas interface. As a result, the fluid dynamics of drops
and menisci can be described taking into account an influence of gravitational
forces and flows on the surface shape. This gives a convenient basis for
respective numerical studies. Estimations of the effects are presented for the
case of an evaporating sessile drop.Comment: 5 page
Whitham systems and deformations
We consider the deformations of Whitham systems including the "dispersion
terms" and having the form of Dubrovin-Zhang deformations of Frobenius
manifolds. The procedure is connected with B.A. Dubrovin problem of
deformations of Frobenius manifolds corresponding to the Whitham systems of
integrable hierarchies. Under some non-degeneracy requirements we suggest a
general scheme of the deformation of the hyperbolic Whitham systems using the
initial non-linear system. The general form of the deformed Whitham system
coincides with the form of the "low-dispersion" asymptotic expansions used by
B.A. Dubrovin and Y. Zhang in the theory of deformations of Frobenius
manifolds.Comment: 27 pages, Late
Multiple addition theorem for discrete and continuous nonlinear problems
The addition relation for the Riemann theta functions and for its limits,
which lead to the appearance of exponential functions in soliton type equations
is discussed. The presented form of addition property resolves itself to the
factorization of N-tuple product of the shifted functions and it seems to be
useful for analysis of soliton type continuous and discrete processes in the
N+1 space-time. A close relation with the natural generalization of bi- and
tri-linear operators into multiple linear operators concludes the paper.Comment: 9 page
Integrable magnetic geodesic flows on Lie groups
Right-invariant geodesic flows on manifolds of Lie groups associated with
2-cocycles of corresponding Lie algebras are discussed. Algebra of integrals of
motion for magnetic geodesic flows is considered and necessary and sufficient
condition of integrability in quadratures is formulated. Canonic forms for
2-cocycles of all 4-dimensional Lie algebras are given and integrable cases
among them are separated.Comment: 16 page
Topological structure of the many vortices solution in Jackiw-Pi model
We construct an M-solitons solutions in Jackiw-Pi model depends on 5M
parameters(two positions, one scale, one phase per solition and one charge of
each solution). By using \phi -mapping method, we discuss the topological
structure of the self-duality solution in Jackiw-Pi model in terms of gauge
potential decomposition. We set up relationship between Chern-Simons vortices
solution and topological number which is determined by Hopf indices and and
Brouwer degrees. We also give the quantization of flux in this case.Comment: 14 pages, 4 figure
Dispersionful analogues of Benney's equations and -wave systems
We recall Krichever's construction of additional flows to Benney's hierarchy,
attached to poles at finite distance of the Lax operator. Then we construct a
``dispersionful'' analogue of this hierarchy, in which the role of poles at
finite distance is played by Miura fields. We connect this hierarchy with
-wave systems, and prove several facts about the latter (Lax representation,
Chern-Simons-type Lagrangian, connection with Liouville equation,
-functions).Comment: 12 pages, latex, no figure
Characterization of unwanted noise in realistic cavities
The problem of the description of absorption and scattering losses in high-Q
cavities is studied. The considerations are based on quantum noise theories,
hence the unwanted noise associated with scattering and absorption is taken
into account by introduction of additional damping and noise terms in the
quantum Langevin equations and input--output relations. Completeness conditions
for the description of the cavity models obtained in this way are studied and
corresponding replacement schemes are discussed.Comment: Contribution to XI International Conference on Quantum Optics, Minsk,
Belarus, 26-31 May, 200
Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type
We consider the special type of the field-theoretical Symplectic structures
called weakly nonlocal. The structures of this type are in particular very
common for the integrable systems like KdV or NLS. We introduce here the
special class of the weakly nonlocal Symplectic structures which we call the
weakly nonlocal Symplectic structures of Hydrodynamic Type. We investigate then
the connection of such structures with the Whitham averaging method and propose
the procedure of "averaging" of the weakly nonlocal Symplectic structures. The
averaging procedure gives the weakly nonlocal Symplectic Structure of
Hydrodynamic Type for the corresponding Whitham system. The procedure gives
also the "action variables" corresponding to the wave numbers of -phase
solutions of initial system which give the additional conservation laws for the
Whitham system.Comment: 64 pages, Late
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